Some Questions Relating to the Age Dynamics of Boreal Forests

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W O R K I N G P A P E R

SOME QUESIlONS

RELATING

TO

THE

AGE DYNAMICS OF B O R W FORESTS

H.H. Shugart M.Ya. Antonovsky

i n t e r n a t i o n a l l n s t ~ t u t e for Appl~ed Systems Analysis

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SOWE QUESITONS RELATING TO THE

AGE

DYNAMICS OF BOREAL FORESTS

H.H.

S h u g a r t M.Ya. Antonousky

July 1988 WP-88-50

Working Papers are interim r e p o r t s on work of t h e I n t e r n a t i o n a l I n s t i t u t e f o r Applied Systems Analysis a n d h a v e r e c e i v e d only limited review. Views o r opinions e x p r e s s e d h e r e i n d o n o t n e c e s s a r i l y r e p r e s e n t t h o s e of t h e I n s t i t u t e o r of i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria

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Foreword

This manuscript i s a r e s u l t of discussions p r i o r t o a n d d u r i n g t h e workshops 'Tmpacts of Change in Climate a n d Atmospheric Chemistry o n N o r t h e r n F o r e s t Ecosystems a n d T h e i r Boundaries" (August 1987) a n d "Global Vegetation Change"

(April 1988) and is a n initial s t e p in t h e development of a s y n t h e s i s between r e a l i s - t i c (e.g. biological-detail-rich) c o m p u t e r o r i e n t e d models of f o r e s t a n d more mathematically, t r a c t a b l e , b u t simpler f o r e s t models. The work i s focused on t h e b o r e a l f o r e s t s of t h e world (an important c a r b o n r e s e r v o i r a n d a n i m p o r t a n t r e s e r v e of softwood timber). The b o r e a l f o r e s t s are a l s o potentially s t r o n g impact systems u n d e r c u r r e n t s c e n a r i o s of COz-induced climate warming.

One p u r p o s e of building a model i s t o g e t a n understanding of what may h a p p e n t o t h e climate if, f o r example, a l l of t h e b o r e a l b e l t were t o d i s a p p e a r , o r if i t s functional efficiency w e r e t o double. Could s u c h a d i s a p p e a r a n c e o c c u r simultaneously with c h a n g e s in t h e t r o p i c a l f o r e s t s ? How would t h i s c h a n g e t h e e x c h a n g e between a t m o s p h e r e a n d t h e e a r t h s u r f a c e ? The a u t h o r s t r y t o d e s c r i b e a f o r e s t ( o r vegetation as a whole) as a boundary l a y e r between f a s t a t m o s p h e r i c p r o c e s s e s a n d slow p r o c e s s e s in soil a n d underground water systems, a n d c o n s i d e r t h e geometry of c a n o p i e s a n d r o o t s as a function of e x t r e m e s c o r r e s p o n d i n g t o a s t a b l e equilibrium of soil a n d underground w a t e r systems.

The a u t h o r s h o p e t o c o n s i d e r t h e s e a n d similar problems d u r i n g t h e i r continu- ing c o o p e r a t i o n .

R.E. Munn L e a d e r

Environment P r o g r a m

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SOME QUESIXONS RELATING

TO THE

AGE DYNAYUCS

OF BOREAL

H.H.

Shugart and M.Ya. Antonovsky

1. MTRODUCTION

P r e s e n t discussions of t h e dynamics of t h e biosphere and of global ecology come

at

a time when t h e r e i s renewed i n t e r e s t in time- and space-scales in ecological systems. An a p p r e c i a t i o n of scales i s a p r e r e q u i s i t e

to

unifying t h e dynamics of t h e atmospheric and oceanographic p r o c e s s with t h e dynamics of t h e t e r r e s t r i a l s u r f a c e . Of p a r t i c u l a r importance i s a knowledge of t h e p a t t e r n s of dominance (in t h e s e n s e of a controlling p a t t e r n ) of p a r t i c u l a r phenomena

at

p a r t i c u l a r scales.

The categorization of c e r t a i n phenomena as being important to understanding t h e s p a c e and time scale in a p a r t i c u l a r ecosystem h a s been t h e topic of reviews f o r s e v e r a l d i f f e r e n t ecosystems (Delcourt

et

al., 1983; O'Neill et al.. 1985). A focus on expressing r e l e v a n t mathematical developments in a manner t h a t

can

provide in- sight into t h e ways ecoystems are s t r u c t u r e d would b e a useful addition

to

t h e s e discussions.

W e are p a r t i c u l a r l y concerned with t h e ecological modeling of t h e world's boreal f o r e s t b e l t a n d w e would posit s e v e r a l r e a s o n s f o r t h i s c o n c e r n .

2. ENVIRONMENTAL CONTROLS OF STAND DYNAMICS

M

BOBeAL FOREST ECOSYSTEMS

The b o r e a l f o r e s t s of t h e world are a major r e p o s i t o r y of t h e world's

terres-

t r i a l o r g a n i c c a r b o n (Bolin, 1986). Moreover, t h e amplitude in t h e annual sinusoidal dynamics of atmospheric c a r b o n dioxide is g r e a t e s t in t h e n o r t h e r n boreal latitudes, and at t h e s e latitudes, t h e r e i s a s t r o n g c o r r e l a t i o n between t h e dynamics of atmospheric c a r b o n dioxide and t h e seasonal dynamics of t h e "green- ness" (Goward

et

al.. 1985) of t h e e a r t h (Tucker

et

^al.,1986). The association

at

h i g h e r n o r t h e r n latitudes of dynamics of atmospheric c a r b o n dioxide and t h e dynamics of

an

index of t h e productivity of t h e vegetation (Tucker

et

al.. 1986) i s c o r r e l a t i o n a l , b u t a possible causal relation (with t h e dynamics of t h e f o r e s t s

at

t h e s e latitudes driving t h e atmospheric c a r b o n concentrations) a p p e a r s

to

b e con- s i s t e n t with t h e p r e s e n t 4 a y understanding of ecological p r o c e s s e s in t h e s e ecosystems (Fung and Tucker, 198?b). Along with i t s familiar role i n plant pho- tosynthesis, C02 i s a "greenhouse" g a s t h a t h a s a n a c t i v e role in governing t h e h e a t budget of t h e e a r t h (Flohn, 1980, Manabe and Stouffer, 1980; Budyko, 1982).

Thus, t h e possibility t h a t t h e b o r e a l f o r e s t s of e a r t h may b e actively participating in t h e dynamics of a n important atmospheric trace gas i s of considerable signifi- cance.

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C u r r e n t predictions of t h e climatic response

to

elevated COz concentrations in t h e atmosphere, motivated by t h e r e c o r d e d i n c r e a s e in atmospheric COz r e p o r t - e d f i r s t

at

^MaunaLoa Observatory in Hawaii (Bacastow and Keeling, 1983) and sub- sequently

at

all latitudes, are based on a r a n g e of l a r g e , physically-based climate models (called g e n e r a l circulation models o r "GCM's"; e.g., Manabe and Stouffer, 1980; Hansen

et

al., 1984; Washington and Meehl, 1984). While t h e GCM's v a r y as t o c e r t a i n underlying assumptions, resolution and o t h e r f e a t u r e s . they converge in t h e i r prediction of a global warming with increased atmospheric COz. The d e g r e e of t h i s warming i s most pronounced at t h e higher latitudes (Dickinson, 1986). Thus, t h e e f f e c t of changes in t h e atmospheric concentration of COz would seem

to

b e strongly d i r e c t e d

to

t h e b o r e a l f o r e s t s of t h e world (Bolin, 1977; S h u g a r t

et

al., 1986).

These large-scale forest/environrnent interactions are a motivation

to

a b e t t e r understanding of t h e environmental p r o c e s s e s controlling t h e

structure

^and function of b o r e a l f o r e s t ecosystems. One hypothesis is t h a t t h e s t r u c t u r e and function of taiga f o r e s t s

are

predominantly controlled by soil thermal and moisture regimes promoted by local topography and t h e successional buildup of a thick f o r e s t floor organic mat. This hypothesis w a s developed by r e s e a r c h e r s working in t h e uplands of i n t e r i o r Alaska (Viereck, 1975; Van Cleve, and Viereck, 1981; Van Cleeve a n d Dyrness, 1983a; Viereck and Van Cleve, 1984; Van Cleve e t al., 1986) where slow growing. n u t r i e n t conservative black s p r u c e (Picea mariana) s t a n d s

oc-

cupy t h e least productive, cold, wet, north-facing s i t e s and f a s t growing, nutrient dynamic white s p r u c e (Picea glauca) and hardwood (Populus temuloides, Betula p a p y r i f e r a ) s t a n d s grow on productive, warmer, d r i e r , south-facing slopes. The low soil t e m p e r a t u r e s and high moisture contents found on permafrost-dominated Picea mariana s i t e s are thought t o act as a negative feedback t h a t promotes

m o s s

growth and inhibits decomposition rates s o t h a t o v e r time t h e f o r e s t floor becomes t h e principal r e s e r v o i r of biomass and nutrients.

Historically, t h e complex unravelling of t h e s e s o r t s of ecological interactions w a s evident in t h e e a r l y work of A.S.

Watt

(1925) on beech f o r e s t s a n d e l a b o r a t e d in his now-classic p a p e r on p a t t e r n and p r o c e s s in plant communities (Watt, 1947).

When one inspects Tansely's (1935) original definition of t h e ecosystem

'These ecosystems, as w e may call them, a r e of t h e most various kinds and sizes. They form one category of t h e multitudinous physical systems of t h e universe. which r a n g e from t h e universe as a whole down t o t h e atom."

. . .

"Actually, t h e systems w e isolate mentally are not only included as p a r t s of t h e l a r g e r ones, but they also overlap, interlock and i n t e r a c t with one another,"

one finds t h a t t h e same concepts t h a t one sees in h i e r a r c h y t h e o r y were explicit in t h e original definition of t h e ecosystem. Of course, t h e Watt/Tansely ecosystem paradigm h a s been introduced as a major ecosystem c o n s t r u c t in ecological studies in t h e United States. One conspicuous example of t h e introduction of t h e s e con- c e p t s w a s Whittaker's (1953) review which used t h e

Watt

pattern-and-process paradigm

to

redefine t h e "climax concept" t h a t w a s (and still i s ) a n important con- s t r u c t in American ecology. These same ideas

are

found in ecosystem concepts developed by Bormann and Likens (1979a. 1979b) in t h e i r "shifting-mosaic steady-

state

concept" of t h e ecosystem as

w e l l as

in what h a s been called a "quasi- equilibrium landscape" (Shugart, 1984).

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3. NON-EQUILIBRIUM DYNAMICS OF ECOLOGICAL SYSTEMS

In

ecosystems t h a t are dominated by sessile organisms, t h e temporal dynamics at t h e s c a l e of t h e individual organism are almost by necessity non-equilibrium dynamics. This is most a p p a r e n t in f o r e s t systems where t h e s p a t i a l s c a l e of t h e individual organisms (the canopy t r e e s ) i s relatively large. The s p a c e below a canopy tree h a s reduced light levels and a considerably a l t e r e d microclimate due t o t h e influence of t h e

tree.

These conditions determine t h e species of

trees

t h a t can survive beneath t h e canopy

tree.

Upon t h e death of t h e canopy t r e e , t h e shading i s eliminated and t h e environment i s changed. In cases in which t h e canopy

tree

d i e s violently (e.g. broken by s t r o n g winds), t h e changes in t h e microenvironment

are

extremely a b r u p t . The d e a t h of t h e canopy t r e e initiates

a

scramble f o r dominance among t h e smaller

trees

t h a t were persisting in t h e environment c r e a t e d by t h e canopy

tree

and seedlings t h a t establish themselves in the high-light environment.

Eventually, one of t h e

trees

becomes t h e canopy dominant. The establishment of a new canopy dominant r e p r e s e n t s t h e closure of t h e death/birth/death cycle t h a t

can

be thought of as t h e typical small-scale behavior of a f o r e s t .

In ecosystems o t h e r t h a n f o r e s t s but s t i l l dominated by sessile organisms, o n e would e x p e c t t h e same s o r t s of dynamics. This nonequilibrium behavior

at

fine spatial s c a l e s h a s been noted in a d i v e r s e a r r a y of ecosystems including c o r a l r e e f s (Connel, 1978; Huston, 1979; Pearson, 1981; Colgan, 1983). fouling communities Karlson, 1978, Kay, 1980). r o c k y inter-tidal communities (Sousa, 1979; Paine and Levin, 1981, Taylor and Littler, 1982; Dethier, 1984) and a wide r a n g e of heath- lands (Christensen, 1985).

The ecosystems t h a t are both historically and c u r r e n t l y t h e m o s t studied in this r e g a r d a r e f o r e s t s . For t h i s r e a s o n i t is worthwhile t o e l a b o r a t e t h e details of t h e death/birth/death p r o c e s s in forests. In f o r e s t s , t h e non-equilibrium dynam- i c s a r e quasi-periodic with t h e period corresponding

to

t h e potential longevity of t h e individual organisms. This "cycle" c a n b e modified by a v a r i e t y of f a c t o r s . One important consideration i s t h e manner of death of t h e dominant

tree.

Some t r e e s typically die violently o r catastrophically and t h e attendant a l t e r a t i o n s of environmental conditions

at

t h e f o r e s t floor (and thus t h e effect on t h e regeneration of potential replacements) are v e r y a b r u p t . Typically t h e s e a b r u p t changes a r e associated with exogenous disturbances but t h e r e a r e some species of

trees

t h a t are "suicidal" in t h a t mature trees flower but once and d i e t o r e l e a s e canopy s p a c e t o t h e progeny (Foster, 1977). Some

trees

tend

to

"waste-away" b e f o r e t h e y die s o t h a t t h e changes in t h e microenvironment t h a t t h e y control a r e more continuous.

Some trees tend

to

s n a p

at

t h e crown when

torn

down by winds; o t h e r s are heaved o v e r at t h e r o o t s exposing mineral soil. All of t h e s e m o d e s of d e a t h (and o t h e r s ) influence t h e stochastic r e g e n e r a t i o n s u c c e s s of t h e

trees

t h a t form t h e next generation.

I t i s a n open question as

to

whether m o d e of death or mode of r e g e n e r a t i o n i s t h e s t r o n g e s t determinant of p a t t e r n diversity in f o r e s t s . Both are a t t r i b u t e s of t h e various

tree

species and may b e strongly i n t e r r e l a t e d . One a s p e c t of t h e mortality of canopy t r e e s and t h e associated opening in t h e f o r e s t canopy ("gap formation") i s t h e size of t h e gap t h a t i s created. S e v e r a l a u t h o r s (van b e Pijl, 1972;

Whitmore, 1975; Grubb, 1977; Bazzaz and Pickett, 1980) have discussed s p e c i e s at- t r i b u t e s t h a t are important in differentiating t h e g a p - s i z e r e l a t e d r e g e n e r a t i o n success of various

trees.

The complexity of t h e regeneration p r o c e s s in

trees

^and i t s a p p a r e n t l y stochastic n a t u r e makes i t v e r y difficult

to

hope

to

p r e d i c t t h e suc- c e s s of a n individual t r e e seedling even if one could determine t h e attendant environmental f a c t o r s . Most c u r r e n t reviewers recognize t h i s and tend t o discuss regeneration in t r e e s from a pragmatic view t h a t t h e f a c t o r s influencing t h e establishment of seedlings can b e usefully grouped in b r o a d classes (Kozlowski, 1971a.

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1971b; van d e r Pijl, 1972; Grubb, 1977; Denslow, 1980).

Since t h e time scales of t h e replacement cycle in f o r e s t s in relatively long, tools f o r a b e t t e r understanding of t h e s e difficult-to-measure phenomena a r e mathematical models of f o r e s t s . O u r c u r r e n t r e s e a r c h i n t e r e s t i s

to

develop a fu- sion of t h e simulation-based stand dynamics models ("Gap models") and more analyt- ically t r a c t a b l e demographic models of t r e e populations. These two a p p r o a c h e s will b e discussed below.

Gap Models.

Gap models

are a

s u b s e t of a class of f o r e s t succession models called individual-tree m o d e l s ( M u m . 1974) because t h e m o d e l s follow t h e growth and f a t e of individual trees. The f i r s t model of t h i s g e n r e w a s t h e JABOWA m o d e l developed by Botkin

et

al. (1972); a similar modeling a p p r o a c h h a s been applied

to

s e v e r a l f o r e s t s in d i f f e r e n t p a r t s of t h e world (see C h a p t e r 4 of S h u g a r t , 1984. f o r a review of s e v e r a l of t h e s e applications, a l s o see K e r c h e r and Axelrod, 1984).

Gap models simulate succession by calculating t h e y e a r

to

y e a r changes in di- a m e t e r of e a c h tree on small plots. The plot size i s determined by t h e size of t h e canopy of a single l a r g e individual. F o r e s t succession dynamics are estimated by t h e a v e r a g e behavior of 5 0 t o 100 of t h e s e plots. The growth of e a c h tree i s d e t e r - mined by t h e a v e r a g e competitive influence of t h e neighboring trees on a plot.

Due

to

t h e small size of plots, gap formation events ( t h e removal of canopy t r e e s through mortality) strongly a f f e c t t h e r e s o u r c e availability on

a

plot which in t u r n a f f e c t s

tree

growth.

The e x a c t location of e a c h t r e e i s not used t o compute competition in t h e s e models. T r e e diameters are used t o determine t r e e height, and then simulated leaf area profiles are computed t o devise competition relationships due t o shading.

These models are spatial in t h a t competition i s computed in t h e v e r t i c a l dimension.

T h e r e i s a n implicit assumption t h a t within a plot of a c e r t a i n size t h e horizontal spatial p a t t e r n s of t h e individual plants do not a f f e c t t h e d e g r e e of competitive stress acting on

an

individual t o any significant d e g r e e beyond t h a t accounted f o r by t h e plant's height (i.e.,

tree

biomass and lead area a r e considered

to

b e homo- genously distributed a c r o s s t h e horizontal dimension of t h e simulated plot).

The regeneration of seedlings on

a

plot and t h e i r subsequent growth i s based on t h e silvicultural c h a r a c t e r i s t i c s of each species, including s i t e requirements f o r germination, sprouting potential, s h a d e t o l e r a n c e , growth potential, longevity, and sensitivity

to

environmental f a c t o r s (water and nutrients). Under optimal growth conditions, t h e growth of a

tree

is assumed

to

o c c u r

at

a

rate

t h a t will pro- duce a n individual of maximum r e c o r d e d a g e and diameter. This curvilinear function grows a

tree to

two-thirds of t h i s maximum diameter at one half of i t s a g e under optimal conditions. Modifications reducing t h i s optimal growth are imposed on e a c h

tree

based on t h e availability of light and, depending on t h e specific model, o t h e r resources. In most gap models, tree growth slows as t h e simulated plot biomass a p p r o a c h e s some maximum potential biomass o b s e r v e d f o r stands of t h e given f o r e s t type. Growth i s f u r t h e r reduced as climate stochastically varies.

Death of an individual t r e e ' s d e a t h i s

a

stochastic process. The probability of a n individual

tree's

death in a given y e a r i s inversely related

to

individual's growth and t h e longevity of its species.

Gap model dynamics are based on information concerning t h e demography and growth of

trees

during t h e lifespan of species. The models have a capability

to

p r e d i c t t h e sequence of replacement of species through time and o t h e r dynamics on t h e scale of t h e a v e r a g e

tree

generation time (Figure 1). A t t h i s s c a l e , t h e suc- c e s s of a tree

at

growing into t h e canopy i s more r e l a t e d t o t h e opportunity f o r

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inseeding into a plot and t h e r e l a t i v e growth r a t e compared

to

o t h e r seedlings than i t i s r e l a t e d

to

t h e distribution of distances from o t h e r competing individuals.

The relationship of t h e height of t h e individual

to

t h e distribution of heights of competitors i s assumed

to

b e sufficient

to

determine t h e level of competitive

stress

experienced by a n individual in relation

to

o t h e r

trees

on t h e plot. This im- plies t h a t t h e distance of a

tree to

i t s competitors h a s no significant influence on t h e amount of light and o t h e r

resources

^available

to

a given tree. In t e r m s of im- plementing t h e s e models, t h e s e assumptions lead

to

a requirement t h a t t h e dynam- i c s of

a

l a r g e number of plots b e averaged

to

b e t t e r estimate t h e mean

rate

of suc- c e s s of canopy invasion of each species.

Because

regeneration, growth and death are modeled on a p r e - t r e e basis and t h e silvics of individuals vary among species, gap m o d e l s are p a r t i c u l a r l y useful

tools

f o r exploring t h e dynamics of mixed-aged and mixed-species f o r e s t s . The models h a v e been tested and validated against independent d a t a (Shugart, 1984, Chapter 4). For t h e s e r e a s o n s , gap m o d e l s c a n also b e w e d

to

e x p l o r e t h e o r i e s about p a t t e r n s in f o r e s t dynamics

at

time scales t h a t are sufficiently long

to

prohi- b i t d i r e c t d a t a collection. Such applications h a v e been instrumental in developing a t h e o r e t i c a l basis f o r understanding t h e coupled e f f e c t s of t r e e d e a t h and regeneration in f o r e s t systems (Shugart, 1984).

One gap model t h a t h a s been used in a l a r g e number of applications in complex, mixed-species, mixed-aged f o r e s t s i s t h e FORET model, a d e r i v a t i v e of t h e JABOWA model (Botkin e t al., 1972). The JABOWA/FORET modeling a p p r o a c h h a s been t h e c e n t r a l topic of two books on t h e dynamics of n a t u r a l f o r e s t s (Bormann and Likens, 1979a; S h u g a r t , 1984). The FORET model and o t h e r analogous models have been modified and applied t o simulate t h e dynamics of a wide r a n g e of f o r e s t s : mixed hardwood f o r e s t s of Australia (Shugart and Noble, 1981); upland f o r e s t of Southern Arkansas (Shugart, 1984); e a s t e r n Canadian mixed species f o r e s t (El- Bayoumi

et

al., 1984); t h e a r i d western coniferous f o r e s t ( K e r c h e r and Axelrod, 1984); a

western

coniferous f o r e s t (Reed and Clark, 1979); and n o r t h e r n hardwood f o r e s t s (Botkin

et

al., 1972; Aber

et

al., 1978, 1979; P a s t o r and P o s t , 1985).

4. DEHOGRAPHICAL YODEL

This t y p e of f o r e s t model r e p r e s e n t s t h e classic a p p r o a c h

to

t h e investigation and prognosis demand of f o r e s t dynamics. I t would a p p e a r

to

us t h a t t h e analytical possibilities of t h i s modeling technique are not completely revealed. The basis of any demographical f o r e s t m o d e l consists of s o m e s e t of dynamical equations f o r t r e e numbers of definite subgroups inside a whole population and f o r some individual

tree

variables

-

masses, lead and

root

s u r f a c e , diameter, etc. The subdivision into subgroups i s dictated by t h e t a s k under consideration and c a n b e v e r y detailed. I t i s usual in classical mathematical ecology

to

d e s c r i b e t h e basic demographical p r o c e s s e s

-

b i r t h , migration, growth and d e a t h

-

by means of definite functions which are derived from t h e o r e t i c a l ideas and empirical data. The poten- t i a l complexity and diversity of dynamical and p a r a m e t e r behavior t h a t t h e corresponding equations demonstrate are unlimited. For example, modeling tech- niques c a n d e s c r i b e systems with many s t a t i o n a r y s t a t e s , oscillations, hysteresis, wave phenomena, stochastic (even chaotic) and adaptive behavior.

W e are c e r t a i n t h a t t h e basic dynamical e f f e c t s shown by gap models in various c o n c r e t e applications (multi-stationary

state

phenomena, t h e s o r t i n g o u t of some

tree

species. when a p p r o p r i a t e conditions v a r y ,

etc.

⁾may b e achieved inside dynamical f o r e s t m o d e l s with r a t h e r simple growth, b i r t h and d e a t h functions and t h e i r dependence upon ecological p a r a m e t e r s in t h e r i g h t p a r t s of equations.

Dynamical m o d e l s give a smooth t r a j e c t o r y ; t h e corresponding t r a j e c t o r y f o r t h e

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gap-model i s obtained by means of a n averaging p r o c e d u r e applied

to

individual gap t r a j e c t o r i e s . So, i t would a p p e a r

to

us t h a t t h e d i f f e r e n c e between t h e

two

modeling a p p r o a c h e s i s not v e r y l a r g e . The gap-model i s applied

to

a v e r a g e

tree

positions inside t h e g a p and determines the f a t e of e a c h

tree

by means of t h e Monte-Carlo mechanism. Dynamical equations are applied

to

a v e r a g e

tree

positions in l a r g e (potentially infinite) t e r r i t o r i e s

to

determine t h e f a t e of definite groups of

trees

by means of viability functions. One c a n match t h e

two

t y p e s of models comparing o n e

tree

in t h e gap with o n e group from a l a r g e t e r r i t o r y and averaging t h e Monte-Carlo variability

to

g e t a determinate variability function. A possible and r a t h e r interesting t a s k of t h i s t y p e h a s not y e t b e e n comprehensively undertaken.

W e see

some

advantages of dynamical models as compared

to

gap-models which h a v e a simulative nature. Firstly, t h e y give t h e possibility of analytical investigation of simple, preliminary models aimed

at

qualitative system analysis. This level of f o r e s t dynamics investigation e n a b l e s o n e

to

d i s c o v e r some b a s i c p r o p e r t i e s of t h e system. F o r example, Antonovsky and Korzukhin (1986), d e s c r i b e d the basic dynamical e f f e c t s of a n e v e n a g e d forest stand by means of

t w o

dynamical v a r i a b l e s ( t r e e number a n d individual

tree

biomass). This model may help

to

make estimations of climatically induced s h i f t s in t h e g e n e r a l system c h a r a c t e r i s t i c s (total biomass, a v e r a g e diameter,

tree

number, etc.). Another example i s a phytophaque interaction with a n u n e v e n a g e d one-species

tree

population (Antonovsky, Kuznet- sov, Clark (1987).

Although both models a r e extremely schematic, t h e y seem

to

b e among t h e simplest models allowing complete qualitative analysis of a system in which t h e p r e - d a t o r differentially a t t a c k s various a g e classes of t h e p r e y .

The main qualitative implications from t h e p r e s e n t p a p e r c a n b e formulated in t h e following, t o some e x t e n t metaphorical, form:

1. The p e s t feeding t h e young trees destabilizes t h e f o r e s t ecosystem more t h a n a p e s t feeding upon old trees. Based upon t h i s implication, w e could t r y

to

explain t h e well-known f a c t t h a t in real ecosystems, p e s t s more frequently feed upon old

trees

t h a n o n young trees. i t seems possible t h a t systems in which t h e p e s t feeds o n young trees may b e less s t a b l e and more vulnerable to e x t e r n a l impacts than systems with t h e p e s t feeding o n old trees. P e r h a p s this h a s led

to

t h e elimination of such systems by evolution.

2. An invasion of a small number of p e s t s into a n existing s t a t i o n a r y f o r e s t ecosystem could r e s u l t in intensive oscillations of i t s a g e s t r u c t u r e .

3. The oscillations could b e e i t h e r damping or periodic.

4. Slow changes of environmental p a r a m e t e r s are a b l e to induce a vulnerability of t h e f o r e s t

to

previously unimportant pests.

Let us now outline possible d i r e c t i o n s f o r extending t h e model. I t seems na-

tural to

t a k e into account the following factors:

1. more t h a n

t w o

a g e classes f o r t h e specified t r e e s ;

2. coexistence of more t h a n o n e tree species a f f e c t e d by t h e pest;

3. introduction of more t h a n o n e p e s t s p e c i e s having various i n t e r s p e c i e s relations;

4. t h e role of variables like foliage area which are important f o r t h e description of defoliation e f f e c t of t h e pest;

5. feedback r e l a t i o n s between vegetation, landscape and microclimate.

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Secondly, comparatively low computer expenses f o r solution of t h e dynamical equations give t h e possibility of a n e x a c t p a r a m e t e r definition even f o r realistic, not simplified models. I t i s well-known t h a t many biological and ecological parame-

ters

are hardly measured in field conditions. s o t h e t a s k a r i s e s of t h e i r identifica- tion by means of comparing model and real behavior ( t r a j e c t o r i e s ) . For example, this a p p r o a c h w a s undertaken in quantitative modeling of post-fire succession in West S i b e r i a (Korzukhin, Sedyh

,

Ter-Mikhaelian, 1987, 1988; Antonovski, Ter- Mikhaelian, 1987; Antonovski, Korzukhin, 1986b).

The dynamical equations were essentially nonlinear, and viability functions were constructed with t h e help of t h e developed t h e o r y of tree competition. Age dynamics of two-species ( c e d a r

+

b i r c h ) uneven-aged s t a n d w a s considered o v e r a 200-year period a f t e r c a t a s t r o p h i c f i r e o c c u r r e n c e . Six important p a r a m e t e r s of t h e system

-

two s e e d s immigration intensities and f o u r inter- and intra-specific competition coefficients were determined by means of t h e usual technique of least s q u a r e minimizing. Wave-like a g e dynamics, typical f o r b o r e a l f o r e s t post- c a t a s t r o p h i c successions, were analyzed from t h e mathematical and ecological points of view. These dynamics

are

quite similar

to

one-gap dynamics during a one-life t r e e cycle (Shugart, 1984).

In s p i t e of t h e roughness of t h e model (Antonovski, Ter-Mikhailian, 1987). in o u r opinion t h e main assumptions

to

b e c o r r e c t e d are assuming a single succession line o v e r t h e e n t i r e a r e a and assuming t h a t all stands are of equal size), s o w e a r e not going t o insist on t h e quantitative exactness of p a r a m e t e r estimations.

Nevertheless, t h e following conclusions seems t o b e non-controversial:

1. Boreal f o r e s t s a r e not in a s t a b l e

state

(in t h e s e n s e of stability of a g e s t r u c - t u r e s ) but t h e r e i s a s t a b l e f i r e regime, i.e., f i r e y e a r s in which a small p a r t of t h e t e r r i t o r y i s burned alternating with major f i r e y e a r s o c c u r r i n g irregu- larly; this conclusion a r i s e s f i r s t l y from t h e nonmonotonous s h a p e s of t h e a g e s t r u c t u r e s and secondly from convergence of t h e dynamics of t h a t p a r t of t h e t e r r i t o r y burned p e r y e a r with t h e p a t t e r n described above; t h e r e a f t e r a s t a b l e p a t t e r n i s maintained.

2. The probabilities of burning i n c r e a s e with t h e a g e of t h e f o r e s t . Other a l t e r - native p a t t e r n s of t h e probability v e c t o r r e s u l t in p a t t e r n s of distribution of f r a c t i o n s of area burned p e r y e a r different to t h e observed ones.

3. The deterministic mechanism of auto-coordination of t h e f o r e s t is insufficient t o explain t h e phenomenon of major f i r e s (because such big differences in values of burning probabilities between s t a g e s i s hardly probable); s o t h e r e should b e a combination of auto-coordination and fluctuations of climatic p a r a m e t e r s t h a t a f f e c t f o r e s t dynamics. Simultaneously t h i s f a c t indicates t h e direction of f u t u r e investigations: t o t a k e as a s t a r t i n g point a v e c t o r of burning probabilities of t h e t y p e obtained in o u r m o d e l (i.e. with values increasing with f o r e s t age) and t o add random fluctuations of climatic p a r a m e t e r s in accordance

to

t h e i r s t a t i s t i c a l distributions constructed with t h e help of long-term observations.

In Antonovski, Glebov, Korzukhin (1987) a n attempt w a s made

to

m o d e l in qualitative

t e r n

t h e dynamics of a n e n t i r e f o r e s t and bog ecosystem which includes abiotic and biotic components. The f o r m e r was t h e thickness of t h e p e a t deposit and t h e l a t t e r w a s t h e f r a c t i o n of hygrophytes in t h e total phytomass. The dynam- i c s of t h e s e two variables modeled by formalizing t h e associated ecological mechanism, w a s t h e main line of this r e s e a r c h .

The proposed m o d e l d e s c r i b e s simultaneously t h e p r o c e s s mechanism f o r a n ecosystem and i t s regional setting because i t i s r e f e r e n c e d t o basic t y p e s of ecological conditions t o b e found in t h e chosen area.

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The bog-formation p r o c e e d s in t w o qualitatively d i f f e r e n t phases. The f i r s t is exogenetic in t h a t t h e system develops under t h e impact of exogenous f o r c e d watering which r e d u c e s a e r a t i o n . The c h a r a c t e r i s t i c time of climatically dictated bogging-debogging fluctuations r a n g e s from s e v e r a l

to

200 y e a r s . The horizontal bogging rates are as high as m e t e r s p e r y e a r .

The second p h a s e i s endogenetic in which t h e gradual bogging f l u c t u a t e s relatively l i t t l e

at

a horizontal rate of centimeters p e r y e a r . For t h i s r e a s o n , t h e bogging i s i r r e v e r s i b l e with usual climatic variations (against whose background exo- genesis o c c u r s ) but i s r e v e r s i b l e o v e r l a r g e time s p a n s during which t h e bog ecosystems are influenced by t h e specifics of mire development, i.e., when regional a s p e c t s become important. The p e a t deposit and t h e impermeable horizon may b e said

to

b e t h e "memory" making t h e system stable. The exogenous watering e f f e c t may b e r e d u c e d with p e a t accumulation p r e s e r v e d . This p h a s e c o v e r s t h e remain- ing p a r t of t h e hydromorphic s e r i e s in t h e exogenetic succession of marshy f o r e s t

-

f o r e s t e d bog --, open bog --, lake-and-bog complex.

APPENDIX

Among t h e huge set of models described above w e will now give m o r e detailed information o n t h e basic model FORET.

The s t r u c t u r e of t h e i n n e r

stream

of d a t a and t h e s t r u c t u r e of organization of i n t e r r e l a t i o n of modules of model FORET show t h a t t h e imitation of successional p r o g r e s s of r e g e n e r a t i o n of f o r e s t stand is essentially a realization of computer p r o c e d u r e of t h e system of equations. In t h e model FORET, t h e y e a r l y i n c r e a s e in t h e diameter of a

tree

i s defined by t h e expression ( f o r notation see S h u g a r t , 1984)

A,Di =BIOM(t). Dmrr,(t) . S M W i ( t ) . I j ( t ) . r n * ( t ) .

(*I

This system of equations i s completed by t h e equations of functional dependence of values of t h e seeking v a r i a b l e from o t h e r variables of t h e system and a l s o by t h e c o n s t r a i n t s t h a t w e put on values of variables of t h e system, for example, z e r o i n c r e a s e in diameter in t h e case of a cold winter.

The simulation of successional p r o c e s s e s in t h e model is realized by using t h e r e s u l t s of studies of t h e life cycle of a tree: b i r t h , growth and death. The recon- struction of a gap in a forest s t a n d is s u p p o r t e d by module PLOTIN (Shugart, 1984).

By t h e end of a simulation of o n e life cycle, t h e existence of gaps i s determined. If such a g a p exists, t h e n t h e model

starts

up again with module PLOTIN until all gaps are filled.

The f e a t u r e of t h e given system of equations is t h e time dependence of a number of equations in t h i s system. The set of equations i s subdivided into t w o s u b

sets

having t h e numbers Nl(t -1) and Nz(t -1) i.e, (N(t -1)

=

Nl(t -1)

+

N2(t -1).

The f i r s t s u b s e t consists of equations whose solution up

to

time

t

d o not involve d i s t u r b a n c e of t h e condition of intersection of t h e lower bound of t h e interval of t h e permissible value of bD(t); t h e second s u b s e t contains all cases in which such a disturbance h a s taken place. Each of t h e s e t w o s u b s e t s is, in t u r n , subdivided on j mutually non-intersecting s u b s e t s of equations, t h e solution of which are defined by a set of values of p a r a m e t e r s c h a r a c t e r i z e d f o r e a c h subset. In o u r

case

t h i s c o r r e s p o n d s

to

a subdivision of t h e modelled f o r e s t stand into s e p a r a t e species.

S o we have

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Selection of excluded equations b e f o r e s t a r t i n g computation

at

^time

t

i s realized with some probability plj f o r e a c h subset and inclusion of equations of t h e second-type i s realized with probability p2. Moreover, from t h e logic of t h e program organization of t h e p r o c e d u r e f o r excluding equations from t h e system, i t fol- lows t h a t t h e formal model f o r exclusion is:

where N-(t) i s t h e number of excluding equations to t h e beginning of calculation f o r t h e moment of time t . Thus, up to time t , t h e r e exist N ( t ) of equations:

The formalization of function ~ + ( t ) is a complex independent problem. The b e s t way t o d e s c r i b e i t as a n algorithm i s through BIRTH (Shugart, 1984). S o t h e complete system of equations are (Trushin, 1986):

1.27+0.3

.

(1-tl(t)13

o . i + o . i

.

( ~ - t ~ ( t ) ) ~ Choice of expression depends on ~ + ( t )

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vij"P2j -v3j.Ijjt'Xjt ( t

u n d e r t h e complex of conditions flji ( 1 )

The modelled value of growth

at

e a c h moment of time is proportional

to

t h e function Xji , defining t h e i n c r e a s e of tree diameter in optimal environmental conditions. The functions rpij(t) and I j i ( t ) define t h e d e c r e a s e in t h i s optimal value of growth due t o competition with o t h e r trees f o r soil n u t r i e n t s and light energy. The functions qzj

( t )

and define t h e d e c r e a s e of "real" growth as a consequence of changes in e x t e r n a l factors such as a i r t e m p e r a t u r e and soil humidity.

A s mentioned previously, t h e model FORET i s devoted t o simulation of t h e process of forming a mature f o r e s t stand in some physio-geographical setting through r e g e n e r a t i o n of t h e f o r e s t s t a n d via filling up t h e f o r e s t gaps. I t i s clear from t h e system of equations how

to

include anthopogenic pollution and how to develop FORET in o t h e r directions.

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